Solution. Because you are not doing business in the state of New York, you only need to do the calculations at the end of each policy year.

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1 Exercises in life and annuity valuation. You are the valuation actuary for the Hard Knocks Life Insurance Company offering life annuities and life insurance to guinea pigs. The valuation interest rate is 2%, and mortality of guinea pigs follows De Moivre s Law with l x = 4 x, 0 x 4. You can assume that times after age 4 are ignored. Your company issues fixed deferred annuities to guinea pigs. The interest rate is guaranteed at 3% for the first two years, and % thereafter. The surrender charge is 0% the first year, 5% the second year, and 0% thereafter. Calculate the reserve for a $000 single premium deferred annuity at issue sold to a newborn. There are no front-end loads. Assume that you use Commissioners Annuity Reserve Valuation Method and you are not doing this in the state of New York. Regulations require that surplus must be 3% of assets at the policy issue date. Find the additional surplus investment requirement for this policy. A. 5 B. 47 C. 44 D. 4 E. 38 Because you are not doing business in the state of New York, you only need to do the calculations at the end of each policy year. Time Fund Value Cash Surrender Value PV of Cash Value As expected, the reserve is given by the present value of the guaranteed cash value at the end of the second policy year, when the surrender charge wears off, and it equals Without a surplus contribution, the company only has 000 worth of assets, and would not be allowed to issue the policy. If the minimum surplus investment is x, then we must have x = 0.03, x and x = = Answer D. 2. You are the valuation actuary for the Hard Knocks Life Insurance Company offering life annuities and life insurance to guinea pigs. The valuation interest rate is 2%, and mortality of guinea pigs follows De Moivre s Law with l x = 4 x, 0 x 4. You can assume that times after age 4 are ignored. Your company issues fixed deferred annuities to guinea pigs. The interest rate is guaranteed at 3% for the first two years, and %

2 thereafter. The surrender charge is 0% the first year, 5% the second year, and 0% thereafter. Calculate the reserve for a $000 single premium deferred annuity at issue sold to a newborn. There are no front-end loads. You are using the Commissioners Annuity Reserve Valuation Method the state of New York. Find required surplus investment if the regulators require 3% surplus at issue for this policy. A. 38 B. 4 C. 44 D. 47 E. 5 The state of New York requires CARVM reserves for all possible dates in the applicable future. Here, the significant dates are first days of each policy year, when previous year s surrender charge wears off, and cash value suddenly increases. These are the numbers for first days of policy years: Time Fund Value Cash Surrender Value PV of Cash Value Note that beyond t = 2, the guaranteed accumulation rate is less than the valuation rate, so those times cannot produce greater present value than Thus the initial reserve is If the capital investment is x, then x = 0.03, x so that x = = Answer E. 3. Your company issues a $,000 fully discrete whole life insurance policy to a population of guinea pigs whose mortality follows De Moivre s Law with l x = 4 x, 0 x 4. Calculate the level benefit premium reserve for such a policy at duration, if issued to a newborn. Assume that the valuation interest rate for this problem is 0%. A. 0 B. 50 C. 00 D. 50 E. 200 Because DeMoivre s Law holds, a newborn guinea pig has a 0.25 probability of dying in each of the years, 2, 3, 4. Thus: A 0 = = (obviously, this is for 0% interest always, because everyone dies eventually),

3 a 0 = = 2.5, 4 Also, P 0 = A 0 a 0 = 2.5 = A = =, a = = 2, 3 and the level premium benefit reserve for this policy is: 000 V 0 = 000A 000P 0 a = = 200. Answer E. 4. Your company issues a $,000 fully discrete whole life insurance policy to a population of guinea pigs whose mortality follows De Moivre s Law with l x = 4 x, 0 x 4. Calculate the level benefit premium reserve for such a policy at duration, if issued to a newborn. Assume that the valuation interest rate for this problem is 0%. Life insurance regulators allow you to use Full Preliminary Term reserving method for this life insurance policy for guinea pigs. What will be the difference between Full Preliminary Term reserve and Level Benefit Premium reserve at time t = 2? A. 0 B. 50 C. 00 D. 50 E. 200 Because DeMoivre s Law holds, a newborn guinea pig has a 0.25 probability of dying in each of the years, 2, 3, 4. Thus: A 0 = = (obviously, this is for 0% interest always, because everyone dies eventually), a 0 = = 2.5, 4 P 0 = A 0 = a = Under Full Preliminary Term, modified benefit premiums will be: First year: α = A 0: = 4, Renewal: β = A a = 2. We have A 2 =, a 2 = + =.5, and therefore V FPT 0 = 000A 2 000β a 2 = = 250.

4 000 V = 000A 000P a = = 400. The modified reserve is significantly smaller, by $50. Answer D. 5. Your company issues a $,000 fully discrete whole life insurance policy to a population of guinea pigs whose mortality follows De Moivre s Law with l x = 4 x, 0 x 4. Assume the 980 National Association of Insurance Commissioners Standard Nonforfeiture Law applies and calculate the minimum surrender cash value required by law for this policy at the end of policy year. Assume that the level benefit premium is used for valuation, and the policy is issued to a newborn. Valuation rate is 0%. A. 52 B. 77 C. 200 D. 233 E. 280 This requires establishing the nonforfeiture premium as prescribed by law. The rule starts with comparison of the benefit premium with Here, level benefit premium is 0.40 (per dollar of benefit), clearly greater than In this case, first year expense allowance for the purpose of nonforfeiture benefits calculation is Note that A 0 = and a 0 = = 2.5. The nonforfeiture renewal premium (per dollar) is: P a = A =.06 a = The nonforfeiture value at time is calculated using basic reserve methodology: CV = 000A 000Pa a = = 52. Answer A. 6. Society of Actuaries Course 50 Sample Examination , Problem No. 30 For a fully discrete whole life insurance of 000 on ( x), you are given: (i) i = (ii) Reserves are determined using a modified reserve method, where the modification period is the entire policy period, the modified premium for each of the first three years is 7.72, and modified premiums are level thereafter. (iii) a x:3 = (iv) a x+3 = (v) 3 E x = (vi) 000A x = (vii) 000A x+20 = Calculate the modified reserve at the end of year 20.

5 We have 000A x = α a x:3 + β 3 a x, so that β = 000A x α a x:3 a = 000A x α a x:3 = 3 x 3E x a x+ 3 The modified reserve sought is V Mod x = 000A x+20 β a x+20 = 000A x+20 β A x+20 i + i Society of Actuaries Course 50 Sample Examination , Problem No. 34 You are given (i) At the beginning of year t At the end of year t Liabilities (reserves) 4,800,000 5,000,000 Surplus,200,000,380,000 (ii) For year t Premium Income 800,000 Investment Income unknown Death Benefits 420,000 Withdrawal Benefits 200,000 Commissions 80,000 Other Expenses 220,000 Increase in Reserves unknown (iii) Premiums, expenses, and commissions are paid at the beginning of the year. (iv) Death benefits and withdrawal benefits are paid at the end of the year. Calculate i, the interest rate earned on assets in year t. A. 5.% B. 5.89% C. 6.46% D. 7.% E. 7.69% Assets at the beginning of the year equal to the sum of liabilities and surplus, i.e., 6,000,000. Similarly, 6,380,000 = Assets at the end of the year, but this also equals 6,000,000 + Premiums Commissions Expenses + Investment Income Death Benefits Withdrawal Benefits = 6,000, ,000 80, ,000 + Investment Income 420, ,000 = = 5,880,000 + Investment Income. Therefore, Investment Income = 500,000. The interest rate earned is 500,000/(6,000,000 + Premiums Commissions Expenses) = = 500,000/6,500,000 = 7.69%. Answer E.

6 8. Society of Actuaries Course 50 Sample Examination , Problem No. 35 For a fully discrete insurance of on ( x), you are given: (i) Reserves are determined using a modified reserve method, where the modification period is the entire policy period, the modified premium for the first year, α, is given by α = α FPT, if β FPT α FPT 0.04, β 0.04, otherwise, and modified premiums, β, are level thereafter. (ii) d = (iii) a x = 3. (iv) a x:0 = (v) A x:0 = (vi) A x: = Calculate β. A B C D E We have α FPT = A x: = 0.006, d a x = = 0.22 = A x = α FPT + β FPT ( a x ) = β FPT, and this results in β FPT = Therefore β FPT α FPT = < 0.04, and full preliminary term applies. Hence, β = β FPT = Answer A. 9. Society of Actuaries Course 50 Sample Examination , Problem No. 36 For a fully discrete insurance of on ( x), you are given: (i) Reserves are determined using a modified reserve method, where the modification period is the entire policy period, the modified premium for the first year, α, is given by α = α FPT, if β FPT α FPT 0.04, β 0.04, otherwise, and modified premiums, β, are level thereafter. (ii) d = 0.06.

7 (iii) a x = 3. (iv) a x:0 = (v) A x:0 = (vi) A x: = Calculate the modified reserve at policy duration 0, 0 V x Mod. A B C D E We have α FPT = A x: = 0.006, d a x = = 0.22 = A x = α FPT + β FPT ( a x ) = β FPT, and this results in β FPT = Therefore β FPT α FPT = < 0.04, and full preliminary term applies. Hence, β = β FPT = 0.24 = Using that value, Mod = A x+0 β a x+0 = d a x+0 β a x+0 = 0 V x = ( ) a x+0 = a x+0. We do not have a x+0 but we can find it from this identity a x = a x:0 + 0 E x a x+0, so that a x+0 = a a x x: = E x Finally, 0V x Mod = a x Answer E. 0. Society of Actuaries Course 50 Sample Examination , Problem No. 5 For a fully discrete whole life insurance of on x ( ), you are given: (i) Cash value is k CV = k V x k SC. (ii) The surrender charge is k SC = P x. (iii) 000P x = 9.55.

8 (iv) 000A x+20 = (v) i = (vi) b k is the amount of paid-up insurance available at time k, determined using the equivalence principle. Calculate 000b 20. A. 254 B. 333 C. 498 D. 578 E. 666 We have 20V x = A x+20 P x a x+20 = A x+20 P x A x+20 = d = Furthermore 20CV = 20 V x 20 SC = Finally 000b 20 = CV A x Answer C.. Society of Actuaries Course 50 Sample Examination , Problem No. 37 For a fully discrete insurance of on ( x), you are given: (i) Reserves are determined using a modified reserve method, where the modification period is the entire policy period, the modified premium for the first year, α, is given by α = α FPT, if β FPT α FPT 0.04, β 0.04, otherwise, and modified premiums, β, are level thereafter. (ii) d = (iii) a x = 3. (iv) a x:0 = (v) A x:0 = (vi) A x: = For a fully discrete 0-year endowment insurance of on ( x), calculate β. A. 0.0 B C D E We have

9 A x:0 = d a x:0 = = Furthermore, α FPT = A x: and so that = 0.006, = A x:0 = α FPT + β FPT ( a x:0 ) = β FPT ( 7.55 ), β FPT = Therefore, β FPT α FPT = > Based on this, for this policy, α = β 0.04, and resulting in = A x:0 = α + β a x:0 β = Answer E. ( ) = β β = 7.55β 0.04, 2. Society of Actuaries Course 50 Sample Examination , Problem No. 37 For a fully discrete insurance of on ( x), you are given: (i) Reserves are determined using a modified reserve method, where the modification period is the entire policy period, the modified premium for the first year, α, is given by α = α FPT, if β FPT α FPT 0.04, β 0.04, otherwise, and modified premiums, β, are level thereafter. (ii) d = (iii) a x = 3. (iv) a x:0 = (v) A x:0 = (vi) A x: = For a fully discrete 0-year endowment insurance of on ( x), calculate the modified reserve at policy duration, V Mod x:0. A. 0 B C D E We have A x:0 = d a x:0 = =

10 Furthermore, α FPT = A x: and so that = 0.006, = A x:0 = α FPT + β FPT ( a x:0 ) = β FPT ( 7.55 ), β FPT = Therefore, β FPT α FPT = > Based on this, for this policy, α = β 0.04, and resulting in = A x:0 = α + β a x:0 β = Therefore ( ) = β β = 7.55β 0.04, V Mod x:0 = α ( + i) q ( β 0.04) x = d A x: d p x A = β 0.04 A x: d A x: x: d = Answer B.

11 o On PV basis, NP covers all benefits, so if premiums commence one second from now, our reserve should be zero! o 5V = 00,000 * A 25+5 NP * ä [25]+5 = * = HW: Reserve at time 0. Answer is Example Mrs. Jackson is a 25 year old that buys a $00K level premium 0 year term policy. The valuation rate is 4.5%, what is the NLP reserve at time 5? Solution Step : Calc NP o Level premium, so 00,000 * A 25:0 / ä [25]:0 = Step 2: Calc Reserve o 5V = Face * A 25+5:5 NP * ä [25]+5:5 = * = ACE Manuals Page 95

12 Example Mrs. Jackson is a 25 year old that buys a $00K increasing premium 0 year, 3-pay term policy. The GP starts at $400 and increases $00 each year. The valuation rate is 4.5%, what is the NLP reserve at time? Age 000q x p x ä x:5 000 A x :5 ä x :0 000 A x: 0 ä x: A x :5 ä x 000 A x Solution Step : Calculate NP o Note that we now have non-level premiums! A 25:0 / ä [25]:3 will not be enough. o PVFB 0 / PVGP 0 = 00,000 * A 25:0 / (GP + GP 2 * v * p x + GP 3 * v 2 * 2 p x ) = / ( * / * p x *p x+ / ) = 82.39% Step 2: Calculate Reserve o V = 00,000 * A 25+: % * ( * /.045) o A 25+:9 = (A 25:0 - vq x ) / vp x =.0570 o V = o a timeline and note that NP is not level!!! ACE Manuals Page 96

13 mpe [x] = m EA [x]:n / ä [x]:m mea [x]:n = expense allowance m t V [x]:n = A [x]+t:n-t ( m P [x]:n + EA / ä [x]:m ) * ä [x]+t:m-t Example Mr. Ehr is getting married at 25 years of age and he buys a level premium, 5 year, $00K term policy. What is the modified reserve at time 5 if the total EA = $00. Age 000q x p x ä x:5 000 A x :5 ä x :0 000 A x: 0 ä x: A x :5 ä x 000 A x Step : Calculate NP o Level premium, so NP = Face * A [25]:5 / ä [25]:5 = /.93 = Step 2: Calculate Reserve without EA o 5V = Face * A 25+5:0 NP * ä [25]+5:0 = * = Step 3: Calculate Modified Reserve (with EA) m t VB [x]:n - m tve [x]:n = (00 /ä [25]:5 ) * ä [25]+5:0 = = 27.7 o Example Mr. Ehr is 25 years of age and he buys a level premium, 5 year, $00K term policy. What is the CRVM reserve at time 5? ACE Manuals Page 97 AB [25]+:4 = AND ä [25]+:4 = Age 000q x p x äx:5 000 Ax :5 äx :0 000 Ax: 0 äx: Ax :5 ä x:9 000 A x: 9 ä x 000 Ax Solution o EA FPT = ( / ) /.045 = o EA 20pay = ( / ) /.045 = o EA CRVM = min ( , ) = o Since this EA is the same as the prior EA, the reserve will not change!!! o If non-level DB, then take average of DB 2 through DB 0 o If EA < 0 (increasing GPs), then floor it at 0 o Note that EA are not linked to expenses, but they do allow for lower reserves as if expenses were included in the NP calculation

14 Exercise You are the valuation actuary for the Hard Knocks Life Insurance Company offering life annuities and life insurance to guinea pigs. The valuation interest rate is %, and mortality of guinea pigs follows De Moivre s Law with l x = 4 x, 0 x 4. You can assume that times after age 4 are ignored. Your company issues fixed deferred annuities to guinea pigs. The interest rate is guaranteed at 5% for the first two years, and 0% thereafter. The surrender charge is 0% the first year, 5% the second year, and 0% thereafter. Calculate the reserve for a $000 single premium deferred annuity at issue sold to a newborn. There are no front-end loads. Assume that you use Commissioners Annuity Reserve Valuation Method and you are not doing this in the state of New York. Regulations require that surplus must be 3% of assets at the policy issue date. Find the additional surplus investment requirement for this policy. Because you are not doing business in the state of New York, you only need to do the calculations at the end of each policy year. Time Fund Value Cash Surrender Value PV of Cash Value The reserve is given by the present value of the guaranteed cash value at the end of the third policy year and it equals Without a surplus contribution, the company only has 000 worth of assets, and would not be allowed to issue the policy. If the minimum surplus investment is x, then we must have x = 0.03, x and x =

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